Add bezier curve to vertices along a plane?
You can apply a bezier curve to a plane using the knife project tool.

Heres my process:

As a separate object, make the curve that you want (mine is pretty bad for a door):


Next, covert the curve to a mesh using the "convert to" tool:



Now, select first the curve mesh, and then the plane, go into edit mode, and do knife project:



The knife project does it from your view, so you want to use align view to make sure you are looking at the plane head-on.

Now, you will have the curve cut into your plane: